A New Approach to Cycling in a 2-locus 2-allele Genetic Model
A New Approach to Cycling in a 2-locus 2-allele Genetic Model
复制标题
DOI：
10.1080/1023619031000076551
复制
发表时间：
2003-05
影响因子：
1.1
通讯作者：
R. Sacker;Hubertus F. von Bremen¶
中科院分区：
文献类型：
--
作者：
R. Sacker;Hubertus F. von Bremen¶
在生殖过程中，由于减数分裂阶段的交叉和重组，出现了新的遗传类型。将开发一种简化的生物模型，该模型结合了这一效应和选择的效应。尽管一条染色体可能包含数千个基因，我们将考虑一个由两个遗传位点组成的简化模型，每个遗传位点包含一些基因的两个等位基因。然后将该模型转化为差分方程或映射模型x*=G(x，r)，其中x表示某一无限群体中的基因型的频率分布，x*是下一代的该分布，r是重组参数。对于某一适应度和重组参数的选择，映射表现出几个固定点。当r变化时，由于线性化映射的特征值的共轭对离开单位圆盘，映射的一个不动点失去其稳定性。结果表明，满足所要求的非共振条件和“非线性阻尼”条件，使不动点发生Neimark-Sacker分叉，达到循环或振荡状态。一旦建立了循环轨道，就可以得出结论，种群的遗传变异(随着时间的推移)可以保持不变。这项工作重新表述并证明了艾伦·黑斯廷斯早期的观察结果，使具有更多遗传位点的染色体的处理变得更加直接。
In the reproductive process new genetic types arise due to crossing over and recombination at the meiotic stage. A simplified biological model will be developed which incorporates this effect and the effect of selection. Although a chromosome may contain thousands of genes we will consider a simplified model consisting of two genetic loci, each containing two alleles of some gene. The model will be then turned into a difference equation or mapping model x* = G(x,r) where x represents the frequency distribution of genotypes in a certain infinite population, x* is this distribution one generation later and r is the recombination parameter. For a certain choice of fitness and recombination parameters the mapping exhibits several fixed points. As r is varied one of the fixed points of the mapping loses its stability due to a conjugate pair of eigenvalues of the linearized mapping leaving the unit disk. It is shown that the required non-resonance conditions and “nonlinear damping” condition are satisfied and thus the fixed point undergoes a Neimark–Sacker bifurcation to a cycling or oscillatory state. Once a cycling orbit is established one can conclude that genetic variation (over time) of the population can be maintained. This work reformulates and proves earlier observations of Alan Hastings in a way that makes the treatment of chromosomes with more genetic loci more straightforward.